21 research outputs found
The quasi-bi-Hamiltonian formulation of the Lagrange top
Starting from the tri-Hamiltonian formulation of the Lagrange top in a
six-dimensional phase space, we discuss the possible reductions of the Poisson
tensors, the vector field and its Hamiltonian functions on a four-dimensional
space. We show that the vector field of the Lagrange top possesses, on the
reduced phase space, a quasi-bi-Hamiltonian formulation, which provides a set
of separation variables for the corresponding Hamilton-Jacobi equation.Comment: 12 pages, no figures, LaTeX, to appear in J. Phys. A: Math. Gen.
(March 2002
Quasi-BiHamiltonian Systems and Separability
Two quasi--biHamiltonian systems with three and four degrees of freedom are
presented. These systems are shown to be separable in terms of Nijenhuis
coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with
an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis
coordinates) and its separability is proved.Comment: 10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May
1997
A class of Poisson-Nijenhuis structures on a tangent bundle
Equipping the tangent bundle TQ of a manifold with a symplectic form coming
from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis
structure from a given type (1,1) tensor field J on Q. It is argued that the
complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ,
but plays a crucial role in the construction of a different tensor R, which
appears to be the pullback under the Legendre transform of the lift of J to
co-tangent manifold of Q. We show how this tangent bundle view brings new
insights and is capable also of producing all important results which are known
from previous studies on the cotangent bundle, in the case that Q is equipped
with a Riemannian metric. The present approach further paves the way for future
generalizations.Comment: 22 page
Involutive orbits of non-Noether symmetry groups
We consider set of functions on Poisson manifold related by continues
one-parameter group of transformations. Class of vector fields that produce
involutive families of functions is investigated and relationship between these
vector fields and non-Noether symmetries of Hamiltonian dynamical systems is
outlined. Theory is illustrated with sample models: modified Boussinesq system
and Broer-Kaup system.Comment: LaTeX 2e, 10 pages, no figure
Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy
We discuss the Boussinesq system with stationary, within a general
framework for the analysis of stationary flows of n-Gel'fand-Dickey
hierarchies. We show how a careful use of its bihamiltonian structure can be
used to provide a set of separation coordinates for the corresponding
Hamilton--Jacobi equations.Comment: 20 pages, LaTeX2e, report to NEEDS in Leeds (1998), to be published
in Theor. Math. Phy
Gaudin Models and Bending Flows: a Geometrical Point of View
In this paper we discuss the bihamiltonian formulation of the (rational XXX)
Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued
spins. In particular, we focus on the homogeneous models. We find a pencil of
Poisson brackets that recursively define a complete set of integrals of the
motion, alternative to the set of integrals associated with the 'standard' Lax
representation of the Gaudin model. These integrals, in the case of su(2),
coincide wih the Hamiltonians of the 'bending flows' in the moduli space of
polygons in Euclidean space introduced by Kapovich and Millson. We finally
address the problem of separability of these flows and explicitly find
separation coordinates and separation relations for the r=2 case.Comment: 27 pages, LaTeX with amsmath and amssym
Quantized W-algebra of sl(2,1) and quantum parafermions of U_q(sl(2))
In this paper, we establish the connection between the quantized W-algebra of
and quantum parafermions of that a
shifted product of the two quantum parafermions of
generates the quantized W-algebra of
Singularities of bi-Hamiltonian systems
We study the relationship between singularities of bi-Hamiltonian systems and
algebraic properties of compatible Poisson brackets. As the main tool, we
introduce the notion of linearization of a Poisson pencil. From the algebraic
viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with
a fixed 2-cocycle. In terms of such linearizations, we give a criterion for
non-degeneracy of singular points of bi-Hamiltonian systems and describe their
types
REDUCTION TECHNIQUES FOR INFINITE-DIMENSIONAL HAMILTONIAN-SYSTEMS - SOME IDEAS AND APPLICATIONS
In the language of tensor analysis on differentaible manifolds we present a reduction method of integrability structures, and apply it to recover some well-known hierarchies of integrable nonlinear evolution equations. The main object dealt with in the paper is the Poisson-Nijenhuis manifold, both in global and in local form: both formulations turn out to be useful, either for theoretical developments or for the applications